"... my father went after the bastards that did it, and attacked them. And they locked him up in Azkaban for it. He never said why he'd done it, because if the Ministry had known what Ariana had become, she'd have been locked up in St. Mungo's for good..."

At the age of six, Percival's daughter Ariana was assaulted by three Muggle boys. The incident left her permanently traumatised and unable to control her magic powers. Percival, in a fit of vengeful rage, attacked her assailants. This resulted in his arrest and conviction by the Ministry of Magic. He was sentenced to Azkaban, where he later died. Percival never admitted the reasons for his attack on the Muggles. This was largely due to his fear that if the Ministry found out about his daughter's delicate condition, they would send her to St. Mungo's Hospital for Magical Maladies and Injuries for the rest of her life, effectively locking her away permanently as a hazard to the International Statute of Wizarding Secrecy.

Percival's attack on the Muggle children led many to believe that he was a supporter of the notion of blood purity. However, this was almost certainly untrue as Percival's own wife was a Muggle-bornwitch. However, the Dumbledore family kept its secrets well, and Percival's reputation would remain tarnished even after the death of his 115-year-old son, Albus.

When Albus Dumbledore put on the Peverell Ring, with hopes that using the Resurrection Stone would allow him to see his loved ones once more, one of the people he hoped to see — and apologise to — was his father.

Physical appearance

As a relatively young wizard, Percival was described as "a good-looking man with eyes that seemed to twinkle," like his son, Albus, who, in his own time at Hogwarts, exhibited many of the same traits as his father. Albus had the same twinkling blue eyes as his father and brother, Aberforth.[1]

Relationships

Kendra Dumbledore

Percival's relationship with his wife during their marriage is unknown, but given the false belief of Percival believing in blood purity, they were a happy married couple.

Albus Dumbledore and Aberforth Dumbledore

Percival's relationship with his sons isn't very clear. Percival didn't seem to be held in high regard by Albus, especially for being imprisoned in Azkaban for an attack on three Muggles, but years later, Dumbledore wished to use the Resurrection Stone to bring back his deceased family and apologize to them - even his father.
Like Albus, Aberforth may not have held his father in high regard either for his attack on the Muggles who attacked his sister, Ariana and being imprisoned in Azkaban.

Ariana Dumbledore

Percival's daughter, Ariana

"...my father went after the bastards that did it, and attacked them. And they locked him up in Azkaban for it. He never said why he'd done it, because if the Ministry had known what Ariana had become, she'd have been locked up in St. Mungo's for good... We had to keep her safe and quiet. We moved house, put it about she was ill, and my mother looked after her, and tried to keep her calm and happy."